Quasi-Poisson - Goodness of fit

Question

Can I use likelihood ratio chi-squared tests to test the goodness of fit for quasi-Poisson models?

Answer 1

If I understand your question correctly: no. Goodness-of-fit tests against an unspecified alternative (e.g., testing either the sum of squares of Pearson residuals or the residual deviance against the upper tail of a $\chi^2_{n-p}$ distribution) depend on the assumption of a fixed mean-variance relationship with no adjustable scale parameters. Once you estimate a dispersion parameter (as in quasi-likelihood), you've essentially soaked up all of the lack of fit into a single parameter. You can compute a pseudo-$R^2$ of some sort, but these are not associated with a particular test.

With quasi-likelihood fits, you're in a similar situation to fitting a linear/ordinary least-squares model; there are plenty of things you can do to examine the goodness of fit (e.g. plot residuals vs fitted to look for bias; scale-location plot to look for heteroscedasticity; Cook's distance/leverage plot to look for outliers), and you can if you like test particular aspects of the fit (e.g. can you reject the hypothesis that the residuals are Normally distributed? [although many people consider this a bad idea]), but there is no overall goodness-of-fit test.

You'll notice that many statistical procedures rely on tests of relative goodness of fit (likelihood ratio test, Wald $t$/$F$ test, AIC and other information criteria ...) but they usually start with a rather vague statement that the initial fit (e.g. the most complex one) should be "good enough" ...